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• Slow shutter time
• Short duration flash
• Rotate the camera around the optical axis of the lens

The dancer is only lit by the very short flash burst, the background is lit by continuous ambient lighting for the much longer exposure time as the camera is being rotated around the lens' optical axis.

Slow Shutter I'd start somewhere in the neighborhood of 1/15-1/30 second. Adjust from there as needed. They key to getting shots like this is to practice them beforehand and find what shutter times work for the way you handle the camera.

Short Flash Compared to the shutter time you'll use for a shot like this, pretty much all small, portable shoe mount type flashes are short duration as long as you're not using them in High Speed Sync (Canon)/Auto Focal Plane (Nikon) mode which tends to act like continuous lighting when the shutter is open. Ironically, with a typical shoe mount flash (even when used off camera) the lower the power setting the shorter the duration of the flash which can make it easier to freeze your subject if there is little to no ambient light falling on the subject. With small portable flashes intensity of the light follows the same pattern regardless of the power setting. Power, i.e. the total amount of light created, is reduced by shortening the length of the flash's duration. So lowering the power is the same thing as shortening the duration. Even at full power, most speedlights have a maximum duration of around 1/1000 second, give or take a stop, which is well above the typical shutter sync speed of 1/200-1/250 for many DSLRs. At lowest power most flashes have a shorter duration than most camera's fastest shutter speed. Keep in mind that with a focal plane shutter even a shutter time of 1/8000 second takes about 1/300 second for the slit between the first and second curtains to transit the sensor (or film).

Camera Rotation It takes a little practice to learn how to rotate the camera while keeping the center of the lens pointed in the same direction. How fast you rotate the camera and how long you leave the shutter open determines how long the blur lines from the ambient lights are. If you don't want circles you can move the camera in other motions.

For more about dragging the shutter, please see:
Why does the flash freeze a picture?
When should you use a normal flash vs a second-curtain flash?
Why doesn't a shutter speed of ¹⁄₂₅₀th freeze motion when a flash of that duration does? (particularly this answer addresses the concerns of your question)
Why does flash decrease motion blur?
Photographing a waterfall but also have a person in foreground

• Slow shutter time
• Short duration flash
• Rotate the camera around the optical axis of the lens

The dancer is only lit by the very short flash burst, the background is lit by continuous ambient lighting for the much longer exposure time as the camera is being rotated around the lens' optical axis.

Slow Shutter I'd start somewhere in the neighborhood of 1/15-1/30 second. Adjust from there as needed. They key to getting shots like this is to practice them beforehand and find what shutter times work for the way you handle the camera.

Short Flash Compared to the shutter time you'll use for a shot like this, pretty much all small, portable shoe mount type flashes are short duration as long as you're not using them in High Speed Sync (Canon)/Auto Focal Plane (Nikon) mode which tends to act like continuous lighting when the shutter is open. Ironically, with a typical shoe mount flash (even when used off camera) the lower the power setting the shorter the duration of the flash which can make it easier to freeze your subject if there is little to no ambient light falling on the subject. With small portable flashes intensity of the light follows the same pattern regardless of the power setting. Power, i.e. the total amount of light created, is reduced by shortening the length of the flash's duration. So lowering the power is the same thing as shortening the duration. Even at full power, most speedlights have a maximum duration of around 1/1000 second, give or take a stop, which is well above the typical shutter sync speed of 1/200-1/250 for many DSLRs. At lowest power most flashes have a shorter duration than most camera's fastest shutter speed. Keep in mind that with a focal plane shutter even a shutter time of 1/8000 second takes about 1/300 second for the slit between the first and second curtains to transit the sensor (or film).

Camera Rotation It takes a little practice to learn how to rotate the camera while keeping the center of the lens pointed in the same direction. How fast you rotate the camera and how long you leave the shutter open determines how long the blur lines from the ambient lights are. If you don't want circles you can move the camera in other motions.

For more about dragging the shutter, please see:
Why does the flash freeze a picture?
When should you use a normal flash vs a second-curtain flash?
Why doesn't a shutter speed of ¹⁄₂₅₀th freeze motion when a flash of that duration does? (particularly this answer addresses the concerns of your question)
Why does flash decrease motion blur?
Photographing a waterfall but also have a person in foreground

• Slow shutter time
• Short duration flash
• Rotate the camera around the optical axis of the lens

The dancer is only lit by the very short flash burst, the background is lit by continuous ambient lighting for the much longer exposure time.

Ironically, with a typical shoe mount flash (even when used off camera) the lower the power setting the shorter the duration of the flash and the easier it is to freeze your subject if there is little to no ambient light falling on the subject. With small portable flashes the peak is just as bright regardless of the power setting. Power is reduced by shortening the length of the flash's duration. So lowering the power is the same thing as shortening the duration. Even at full power, most speedlights have a maximum duration of around 1/1000 second which is well above the typical shutter sync speed of 1/200-1/250 for many DSLRs. At lowest power most flashes have a shorter duration than your camera's fastest shutter speed. Keep in mind that with a focal plane shutter even a shutter time of 1/8000 second takes about 1/300 second for the slit between the first and second curtains to transit the sensor (or film).

For more about dragging the shutter, please see:
Why does the flash freeze a picture?
When should you use a normal flash vs a second-curtain flash?
Why doesn't a shutter speed of ¹⁄₂₅₀th freeze motion when a flash of that duration does? (particularly this answer addresses the concerns of your question)
Why does flash decrease motion blur?
Photographing a waterfall but also have a person in foreground

• Slow shutter time
• Short duration flash
• Rotate the camera around the optical axis of the lens

The dancer is only lit by the very short flash burst, the background is lit by continuous ambient lighting for the much longer exposure time as the camera is being rotated around the lens' optical axis.

Slow Shutter I'd start somewhere in the neighborhood of 1/15-1/30 second. Adjust from there as needed. They key to getting shots like this is to practice them beforehand and find what shutter times work for the way you handle the camera.

Short Flash Compared to the shutter time you'll use for a shot like this, pretty much all small, portable shoe mount type flashes are short duration as long as you're not using them in High Speed Sync (Canon)/Auto Focal Plane (Nikon) mode which tends to act like continuous lighting when the shutter is open. Ironically, with a typical shoe mount flash (even when used off camera) the lower the power setting the shorter the duration of the flash which can make it easier to freeze your subject if there is little to no ambient light falling on the subject. With small portable flashes intensity of the light follows the same pattern regardless of the power setting. Power, i.e. the total amount of light created, is reduced by shortening the length of the flash's duration. So lowering the power is the same thing as shortening the duration. Even at full power, most speedlights have a maximum duration of around 1/1000 second, give or take a stop, which is well above the typical shutter sync speed of 1/200-1/250 for many DSLRs. At lowest power most flashes have a shorter duration than most camera's fastest shutter speed. Keep in mind that with a focal plane shutter even a shutter time of 1/8000 second takes about 1/300 second for the slit between the first and second curtains to transit the sensor (or film).

Camera Rotation It takes a little practice to learn how to rotate the camera while keeping the center of the lens pointed in the same direction. How fast you rotate the camera and how long you leave the shutter open determines how long the blur lines from the ambient lights are. If you don't want circles you can move the camera in other motions.

For more about dragging the shutter, please see:
Why does the flash freeze a picture?
When should you use a normal flash vs a second-curtain flash?
Why doesn't a shutter speed of ¹⁄₂₅₀th freeze motion when a flash of that duration does? (particularly this answer addresses the concerns of your question)
Why does flash decrease motion blur?
Photographing a waterfall but also have a person in foreground

• Slow shutter time
• Short duration flash
• Rotate the camera around the optical axis of the lens

The dancer is only lit by the very short flash burst, the background is lit by continuous ambient lighting for the much longer exposure time.

Ironically, with a typical shoe mount flash (even when used off camera) the lower the power setting the shorter the duration of the flash and the easier it is to freeze your subject if there is little to no ambient light falling on the subject. Even at full power, most speedlights have a maximum duration of around 1/1000 second which is well above the typical shutter sync speed of 1/200-1/250 for many DSLRs. At lowest power most flashes have a shorter duration than your camera's fastest shutter speed. Keep in mind that with a focal plane shutter even a shutter time of 1/8000 second takes about 1/300 second for the slit between the first and second curtains to transit the sensor (or film).

For more about dragging the shutter, please see:
Why does the flash freeze a picture?
When should you use a normal flash vs a second-curtain flash?
Why doesn't a shutter speed of ¹⁄₂₅₀th freeze motion when a flash of that duration does? (particularly this answer addresses the concerns of your question)
Why does flash decrease motion blur?
Photographing a waterfall but also have a person in foreground

• Slow shutter time
• Short duration flash
• Rotate the camera around the optical axis of the lens

The dancer is only lit by the very short flash burst, the background is lit by continuous ambient lighting for the much longer exposure time.

Ironically, with a typical shoe mount flash (even when used off camera) the lower the power setting the shorter the duration of the flash and the easier it is to freeze your subject if there is little to no ambient light falling on the subject. With small portable flashes the peak is just as bright regardless of the power setting. Power is reduced by shortening the length of the flash's duration. So lowering the power is the same thing as shortening the duration. Even at full power, most speedlights have a maximum duration of around 1/1000 second which is well above the typical shutter sync speed of 1/200-1/250 for many DSLRs. At lowest power most flashes have a shorter duration than your camera's fastest shutter speed. Keep in mind that with a focal plane shutter even a shutter time of 1/8000 second takes about 1/300 second for the slit between the first and second curtains to transit the sensor (or film).

For more about dragging the shutter, please see:
Why does the flash freeze a picture?
When should you use a normal flash vs a second-curtain flash?
Why doesn't a shutter speed of ¹⁄₂₅₀th freeze motion when a flash of that duration does? (particularly this answer addresses the concerns of your question)
Why does flash decrease motion blur?
Photographing a waterfall but also have a person in foreground